View source for Poisson equation

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A partial differential equation which is satisfied by the [[Potential|potential]] of a mass distribution inside domains occupied by the masses creating this potential. For the [[Newton potential|Newton potential]] in the space $\mathbf R^n$, $n\geq3$, and the [[Logarithmic potential|logarithmic potential]] in $\mathbf R^3$ the Poisson equation has the form

$$\Delta u=\sum_{i=1}^n\frac{\partial^2u}{\partial x_i^2}=-\sigma(S^n)\rho(x_1,\dots,x_n),$$

where $\rho=\rho(x_1,\dots,x_n)$ is the density of the mass distribution, $\sigma(S^n)=n\pi^{n/2}/\Gamma(n/2+1)$ is the area of the unit sphere $S^n$ in $\mathbf R^n$ and $\Gamma(n/2+1)$ is the value of the gamma-function.

Poisson's equation is a basic example of a non-homogeneous equation of elliptic type. The equation was first considered by S. Poisson (1812).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Bitsadze,   "Equations of mathematical physics" , MIR  (1980)  (Translated from Russian)  {{MR|0587310}} {{MR|0581247}} {{ZBL|0499.35002}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Courant,   D. Hilbert,   "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)  {{MR|0195654}} {{ZBL|}} </TD></TR></table>



====Comments====
The map $u\mapsto\Delta(u)$ defines a morphism from the sheaf of local differences of superharmonic functions into a sheaf of measures on $\mathbf R^n$. This remark leads to a treatment of the Poisson problem in the framework of harmonic spaces (cf. [[Harmonic space|Harmonic space]]), see [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.-Y. Maeda,   "Dirichlet integrals on harmonic spaces" , ''Lect. notes in math.'' , '''803''' , Springer  (1980)  {{MR|0576059}} {{ZBL|0426.31001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.D. Poisson,   "Remarques sur une équation qui se présente dans la théorie des attractions des sphéroïdes"  ''Nouveau Bull. Soc. Philomathique de Paris'' , '''3'''  (1813)  pp. 388–392  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Rudin,   "Function theory in the unit ball in $\mathbf C^n$" , Springer  (1980)  {{MR|601594}} {{ZBL|0495.32001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  O.D. Kellogg,   "Foundations of potential theory" , F. Ungar  (1929)  (Re-issue: Springer, 1967)  {{MR|0222317}} {{MR|1522134}} {{ZBL|0152.31301}} {{ZBL|0053.07301}} </TD></TR></table>